Talk:Ballium's number
Can it be computed? I'm not sure that it is possible, using current technology, to actually compute Ballium's number up to the unit's digit. We cannot infer that it is possible based on the fact that pi has been computed to trillions of digits, as they use different algorithms; Chudnovsky's algorithm is all integer operations, plus one square root operation. The question is, how difficult is it to take the power of one transcendental number to another, with an accuracy of 138 billion digits? Deedlit11 (talk) 11:15, May 20, 2013 (UTC) :138 billion digits is really too large amount to my virtual machine (I can't even store them). Ikosarakt1 (talk ^ ) 12:21, May 20, 2013 (UTC) ::138 billion digits is still finite, unlike infinity. Jiawhein \(a\)\(l\) 13:03, May 20, 2013 (UTC) Exact number of digits and first digits I think that the exact number of digits (before the decimal point) should be added to the article. -- 20:04, September 16, 2014 (UTC) :You're welcome to try and do the computation. Deedlit11 (talk) 20:28, September 16, 2014 (UTC) :Oh wait, you just wanted the number of digits. Deedlit11 (talk) 21:01, September 16, 2014 (UTC) :This page would EXPLODE due to billions of digits. So NO. User:Antares.I.G.Harrison (talk) ::No, he/she is just asking the number of digits, which is a 12-digit number and is already added. -- ☁ I want more ⛅ 13:52, February 16, 2015 (UTC) Transcendence Schanuel's conjecture would imply the transcendence of Ballium's number: Firstly, ln(π) and iπ are linearly independent over Q''', so '''Q(ln(π), iπ, π, -1) has transcendence degree of at least 2 over Q'''. Therefore, ln(π) and π are algebraically independent over '''Q. Secondly, ln(π), π, and iπ are linearly independent over Q''', so '''Q(ln(π), π, iπ, π, e^π, -1) has transcendence degree of at least 3 over Q'''. Therefore, ln(π), π, and e^π are algebraically independent over '''Q. Thirdly, ln(π), π, iπ, and ln(π)*e^π are linearly independent over Q''', so '''Q(ln(π), π, iπ, ln(π)*e^π, π, e^π, -1, π^(e^π)) has transcendence degree of at least 4 over Q'''. Therefore, ln(π), π, e^π, and π^(e^π) are algebraically independent over '''Q. And fourthly, ln(π), π, iπ, ln(π)*e^π, and π^(e^π) are linearly independent over Q''', so '''Q(ln(π), π, iπ, ln(π)*e^π, π^(e^π), π, e^π, -1, π^(e^π), e^(π^(e^π))) has transcendence degree of at least 5 over Q'''. Therefore, ln(π), π, e^π, π^(e^π), and e^(π^(e^π)) are algebraically independent over '''Q. It follows that e^(π^(e^π)), and therefore also Ballium's number = (794,843,294,078,147,843,293.7 + 1/30) * e^(π^(e^π)), are transcendental numbers. -- 14:20, September 29, 2014 (UTC) :Nice! But, could you explain how we know that ln(π) and iπ are linearly independent over Q? Deedlit11 (talk) 15:17, September 29, 2014 (UTC) ::Because ln(π) is a nonzero real number, and iπ a nonzero imaginary number. -- 15:21, September 29, 2014 (UTC) :::Derp. Deedlit11 (talk) 15:28, September 29, 2014 (UTC) What does it mean for pi to be independant over Q?Boboris02 (talk) 17:06, October 26, 2016 (UTC) :No one said independant. Alysdexia (talk) 23:09, April 25, 2018 (UTC)